250 Prof. F. Y. Edge worth on the Application of 



interval between any two o£ them *. There are at first 

 supposed to be no external forces. To investigate the 

 ultimate distribution of velocities to which this system 

 tends, three distinct lines of reasoning may be employed. 



I. Among arguments accounting for the genesis of the 

 normal law of error a foremost place seems due to the rationale 

 adopted by Laplace (with reference to averages of obser- 

 vations). On that analogy we may expect that the velocity 

 will be distributed according to the normal law f, if it is a 

 linear function of elements each of which m;iy be regarded 

 as a random specimen of a group distributed according to 

 some definite law of frequency, say %(U, u) — not, in general, 

 the same law for different elements. Consider the play 

 of the system during an interval of time At so short that 

 the deformation of the frequency-function through the 

 occurrence of collisions is very slight. If a piston of 

 mass M with velocity U overtakes one of mass m with 

 velocity u %, the new velocity of the former may be written 



U' = (M-m)U + 2mu; (1) 



if for brevity we take M + ?n as the unit of mass. Now 

 let the cylinder with velocity U make a second collision 

 with one of mass m and velocity u\. We have for the new 

 velocity of the M corpuscle, 



1J'' = (M-m)U' + 2mu 1 = (M.-m)lJ + 2m(M-in)u + 2mi( 1 ; 



• • • (2) 

 and so on. Since the frequency for the distribution of 

 velocities may be treated as constant during At, the 

 collisions tend to bring the distribution nearer to the 

 normal §. If the frequency-function was %(U, u) initially, 

 let it become, in the interval At, p£ + A-%, or % lm By parity, 

 in a second interval there will be a further approach to the 



* If this assumption is not granted, it will be easy to make allowance 

 in the formulae for the length of a piston, supposed small (with reference 

 to the mean free path), though not negligible. 



t By the " normal law '' of distribution (or "error") is here under- 



1 r 2 



stood (conformity to) the function 7 - exp. — — , and the analogous 



V 7TC c 2 



expressions for higher dimensions : e. g. (3) and (5) below. 



\ The frequency distribution is not altered by the collision of like 

 molecules, since they simply exchange velocities. 



§ According to the general Lav) of Error as exhibited by the present 

 writer in the ' Cambridge Philosophical Transactions ' for 1905, vol. xx. 

 Cp. l Journal of the Royal Statistical Society/ Sept. 1906, and ' Philo- 

 sophical Magazine,' 1889, vol. xxviii. p. 283 (article by Dr. Burton). 



